Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)



Secondary Chern-Euler forms and the law of vector fields

Author: Zhaohu Nie
Journal: Proc. Amer. Math. Soc. 140 (2012), 1085-1096
MSC (2000): Primary 57R20, 57R25
Published electronically: July 1, 2011
MathSciNet review: 2869093
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Law of Vector Fields is a term coined by Gottlieb for a relative Poincaré-Hopf theorem. It was first proved by Morse and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner part of its tangential projection on the boundary. We give two elementary differential-geometric proofs of this topological theorem in which secondary Chern-Euler forms naturally play an essential role. In the first proof, the main point is to construct a chain away from some singularities. The second proof employs a study of the secondary Chern-Euler form on the boundary, which may be of independent interest. More precisely, we show by explicitly constructing a primitive that away from the outward and inward unit normal vectors, the secondary Chern-Euler form is exact up to a pullback form. In either case, Stokes' theorem is used to complete the proof.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57R20, 57R25

Retrieve articles in all journals with MSC (2000): 57R20, 57R25

Additional Information

Zhaohu Nie
Affiliation: Department of Mathematics, Penn State Altoona, 3000 Ivyside Park, Altoona, Pennsylvania 16601
Address at time of publication: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322

Received by editor(s): December 15, 2010
Published electronically: July 1, 2011
Communicated by: Jianguo Cao
Article copyright: © Copyright 2011 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia